Notes and course summaries on double integrals, all the rules you need to solve your exercises. The words are in basic English but the math part is easy to grasp.
Multiple integrals refer to the integration of functions of multiple variables over multiple regions of space.
In particular, they involve integrating a function of two or more variables over a two-dimensional region, a
three-dimensional region, or a higher-dimensional region.
For example, the double integral of a function 𝑓(𝑥, 𝑦) over a region ℝ in the 𝑥𝑦-plane can be thought of as
finding the volume under the surface 𝑧 = 𝑓(𝑥, 𝑦) and above the region ℝ. Similarly, the triple integral of a
function 𝑓(𝑥, 𝑦, 𝑧) over a region in three-dimensional space can be thought of as finding the volume under
the surface 𝑠 = 𝑓(𝑥, 𝑦, 𝑧) and above the region.
Multiple integrals are a fundamental tool in calculus, and they have many applications in physics,
engineering, economics, and other fields.
Let’s start with some maths now.
I. Double integral :
Let : 𝑓 : 𝐷 ⟼ ℝ2
(𝑥, 𝑦) ⟼ 𝑧 = 𝑓(𝑥, 𝑦)
a continuous and integrable function on 𝐷.
The double integral is denoted by ∬𝑫 𝒇(𝒙, 𝒚). 𝒅𝒙𝒅𝒚 and physically represents the value of the
volume of the cylinder with base D and ℎ𝑒𝑖𝑔ℎ𝑡 = 1 covered by the surface 𝑧 = 𝑓(𝑥, 𝑦).
▪ Let : 𝐷 = 𝐷1 ∪ 𝐷2 with 𝐷1 ∩ 𝐷2 = 𝑠𝑖𝑚𝑝𝑙𝑒 𝑐𝑢𝑟𝑣𝑒 (i.e, a curve whose points are neither in 𝐷1
nor in 𝐷2 ).
𝐷≡
Stuvia customers have reviewed more than 700,000 summaries. This how you know that you are buying the best documents.
Quick and easy check-out
You can quickly pay through credit card or Stuvia-credit for the summaries. There is no membership needed.
Focus on what matters
Your fellow students write the study notes themselves, which is why the documents are always reliable and up-to-date. This ensures you quickly get to the core!
Frequently asked questions
What do I get when I buy this document?
You get a PDF, available immediately after your purchase. The purchased document is accessible anytime, anywhere and indefinitely through your profile.
Satisfaction guarantee: how does it work?
Our satisfaction guarantee ensures that you always find a study document that suits you well. You fill out a form, and our customer service team takes care of the rest.
Who am I buying these notes from?
Stuvia is a marketplace, so you are not buying this document from us, but from seller scienceexplore. Stuvia facilitates payment to the seller.
Will I be stuck with a subscription?
No, you only buy these notes for $3.25. You're not tied to anything after your purchase.
